if we start from three lines with a common point, then add two lines through some point of that triad, we have degree 5 and only two triple points, so adding 6 more lines gives only 14 triple points. thus even if we can smooth the extraneous double points we lose. maybe one should pass some conics through more nodes. we could pass one conic through all 4 nodes at step 3 above, getting a degree 7 curve with 6 triple points, and 4 nodes. passing another conic through all 4 gives a degree 9 curve with 10 triple points, and 10 nodes. so we get a reducible one, then try to smooth it. /
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roy smithMay 31 '12 at 2:24

According to the paper, the virtual dimension of the linear system is $78-90-1<0$ since the degree of the equation is at least $5$. So it looks like the answer is no.
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Will SawinMay 31 '12 at 3:00

6

Dumitrescu's paper deals with points in general position (even if that may not be clear in the intro). The singular points of that rational curve would certainly not be in general position; that can happen. For instance, there are indeed curves of degree 10 with 12 double points. These two papers by Gradolato-Mezzetti are more relevant: "Families of curves with ordinary singular points on regular surfaces". Ann. Mat. Pura Appl. (4) 150 (1988), 281–298. and "Curves with nodes, cusps and ordinary triple points", Ann. Univ. Ferrara Sez. VII (N.S.) 31 (1985), 23–47 (1986). It's Roy's approach.
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quimMay 31 '12 at 15:22

Yes, the points could be in special position, of course. The simplest example is sextic with 10 double points. A lot of such curves exist, but the points are not in general position. I will try to find the articles you mention. They seem to be very interesting.
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Jérémy BlancJun 13 '12 at 16:55

I found the articles and some rough bounds which give existence of curves with triple points, but the bounds seem to be too far from my explicit question.
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Jérémy BlancJun 25 '12 at 14:57